Video Explanation

Text Explanation

Let’s start with each of the statements (1) and (2) and see if they, alone or combined, can lead us to conclude about the remainder when the positive integer n is divided by 4.

(1) “When n is divided by 8, the remainder is 1”

This means that the integer n − 1 is a multiple of 8 or

 where k an integer.

Introducing 4 in the denominator, this can also be written as

And using cross multiplication:

  which is always an integer (since k is an integer)

So, n − 1 is a multiple of 4 which means that the remainder when the positive integer n is divided by 4 is 1.

Therefore statement (1) lead us to conclude about the statement in the prompt so it is a sufficient statement.

(2) “When n is divided by 2, the remainder is 1”

The fastest way to think about it is to recognize that statement (2) basically says that n is an odd number 1, 3, 5, 7, 9, 11, etc. When an odd number is divided by 4, the possible remainders are 1 or 3 (0 and 2 are not possible because this would mean that the number is even and not odd). So, we cannot conclude about the remainder of the division of n by 4 in this case. This statement is not sufficient.